Wherever ten Jews live, they have a mitzva to designate a place for their prayers. It becomes a small sanctuary (mikdash me’at) and is called a beit knesset (synagogue) (MT, Laws of Prayer 11:1).
It is important to note that a synagogue is not just a place that facilitates prayer with a minyan; rather, it has independent value as a place designated for devarim she-bikdusha, a place where the Shekhina dwells. It is a great mitzva to build a synagogue, as the Torah states: “They shall make Me a sanctuary and I will live among them” (Shemot 25:8). Although this verse primarily comes to teach us about the building of the Temple, an offshoot of that mitzva is the mitzva to build a synagogue (see Peninei Halakha: Collected Essays I, ch. 1 n. 1). This is what the prophet Yeĥezkel meant by “I have been for them a small sanctuary” (Yeĥezkel 11:16), as R. Yitzĥak interpreted, “These are the synagogues and batei midrash (study halls)” (Megilla 29a).
Especially after the destruction of our Temple and our exile from our land – and the exile of the Shekhina along with us – as a result of which we are unable to ascend and appear before the Lord, our God, the synagogues and study halls serve as places in which we can preserve the memory of the Shekhina’s manifestation at the Temple. Likewise, the prayers recited in the synagogue were instituted to correspond to the sacrificial offerings brought at the Temple. Hence, synagogues are constructed for two purposes: to facilitate organized prayer, and to serve as a miniature Temple, a commemoration and extension of the Temple’s sanctity (see Peninei Halakha: Collected Essays I, ch. 6 concerning the laws of the synagogue; here we outline the laws that pertain to women).
Since the idea of the synagogue is so important, all members of the community must participate in the funding of its construction. If there are Jews who are not interested in participating in payments for building the synagogue, the leaders of the community may compel them to contribute its funding (SA 150:1, which states that it was customary to divide the payment according to wealth, though some divided the burden on a per capita basis).
It seems to me that the mitzva to build a synagogue applies to women as well, and it is good that she share in the payments for its construction. Even though women need not pray in the synagogue, they are nevertheless considered members of the community, and thus the mitzva to build a “small sanctuary” for the community applies to them as well. Additionally, it is worth noting that the Israelite women donated their jewelry to the Mishkan. Furthermore, women do come to pray in the synagogue sometimes; as we learned (above, 20:2), there is merit and value in a woman praying in a synagogue. Nonetheless, if a woman does not want to help financially support the building of the synagogue, it seems that she cannot be compelled to do so, since she is not obligated to pray in it.